184. Surds and Decimals

Learning Intentions

• Recognise square roots are irrational.
• Estimate irrational numbers Use nearby perfect squares.
• Compare rational and irrational numbers on a number line.

Pre-requisite Summary

• Know that a rational number can be written as a fraction , where and are integers and .
• Know that terminating and recurring decimals are rational.
• Know common perfect squares such as and .
• Understand that means the positive number that squares to give .
• Know how to place positive and negative numbers on a number line.
• Know that numbers further to the right on a number line are greater.

Worked Examples

Worked Example 1

Decide whether each square root is rational or irrational.

a)

b)

c)

Worked Example 2

Decide whether each statement is true or false.

a) is irrational.

b) is irrational.

c) is rational.

Worked Example 3

Estimate each irrational number by finding the two whole numbers it lies between.

a)

b)

c)

Worked Example 4

Use nearby perfect squares to estimate each value to one decimal place.

a)

b)

c)

Worked Example 5

Place the following numbers in ascending order.

Worked Example 6

Compare each pair of numbers using , or .

a) and

b) and

c) and

Problems

Problem 1

Decide whether each square root is rational or irrational.

a)

b)

c)

Problem 2

Decide whether each statement is true or false.

a) is irrational.

b) is irrational.

c) is rational.

Problem 3

Estimate each irrational number by finding the two whole numbers it lies between.

a)

b)

c)

Problem 4

Use nearby perfect squares to estimate each value to one decimal place.

a)

b)

c)

Problem 5

Place the following numbers in ascending order.

Problem 6

Compare each pair of numbers using , or .

a) and

b) and

c) and

Exercises

Understanding and Fluency

Exercise 1.

Decide whether each square root is rational or irrational.

a)

b)

c)

d)

Exercise 2.

Decide whether each square root is rational or irrational.

a)

b)

c)

d)

Exercise 3.

State the two whole numbers each irrational number lies between.

a)

b)

c)

d)

Exercise 4.

State the two whole numbers each irrational number lies between.

a)

b)

c)

d)

Exercise 5.

Use nearby perfect squares to estimate each value to one decimal place.

a)

b)

c)

d)

Exercise 6.

Use nearby perfect squares to estimate each value to one decimal place.

a)

b)

c)

d)

Exercise 7.

Compare each pair of numbers using , or .

a) and

b) and

c) and

d) and

Exercise 8.

Place each set of numbers in ascending order.

a)

b)

Reasoning

Exercise 9.

Explain why is rational but is irrational.

Exercise 10.

A student says, “Every square root is irrational because square roots usually have long decimals.” Explain why this is incorrect.

Exercise 11.

Without using a calculator, explain why must be between and .

Exercise 12.

A student places to the left of on a number line. Explain the mistake.

Exercise 13.

Decide which number is greater: or . Justify your answer using nearby perfect squares.

Exercise 14.

Explain why is less than without using a calculator.

Problem-solving

Exercise 15.

A number line has the marks and shown. Place approximately on the number line and explain your estimate.

Exercise 16.

Three students estimate .

• Alex:
• Brianna:
• Chen:

Who has the best estimate? Explain your reasoning.

Exercise 17.

Order the following numbers from smallest to largest.

Exercise 18.

A square has an area of . Estimate the side length of the square to one decimal place.

Exercise 19.

A point on a number line is slightly greater than but less than . Give one possible irrational square root that could be placed at this point and justify your choice.

Exercise 20.

Create a set of four numbers containing:

• one rational square root
• one irrational square root between and
• one fraction between and
• one decimal between and

Then place them in ascending order.

Potential Misunderstandings

• Students may think all square roots are irrational, but square roots of perfect squares are rational, such as .

• Students may think a square root is irrational only if it has a decimal form, but irrationality means it cannot be written as a fraction where and are integers and .

• Students may confuse with or think square root means “divide by ”.

• Students may estimate as halfway between and because is halfway between and , but square roots do not increase evenly in this way.

• Students may forget to use nearby perfect squares when estimating irrational square roots.

• Students may place between the wrong whole numbers by comparing directly to the whole numbers instead of comparing to perfect squares.

• Students may compare and by only looking at the number under the square root.

• Students may think irrational numbers cannot be placed on a number line because they cannot be written exactly as terminating decimals.

• Students may incorrectly order rational and irrational numbers when fractions, decimals and square roots appear together.

Next: 185. Ordering Real Numbers